Astro 6531: Astrophysical Fluid Dynamics

Fall 2024

Instructor: Profs. Dong Lai & Steve Lantz

Dong's office: 618 SSB; email: DL57_at_cornell.edu; Steve's office: 614 SSB or 731 Rhodes; email: steve.lantz_at_cornell.edu
Office hours: After class on M & W. Other times are fine; best contact by email, which will usually be answered within 24 hours.

Time & Place: Monday and Wednesday 10:10 am - 11:25 am, SSB 301

Class website: https://donglai6.github.io/a6531.html

Description:

Three-credit lecture course, aimed at general astrophysics/physics/engineering graduate students as well as well-prepared undergraduate students.

A knowledge of fluid dynamics is essential for understanding many of the most interesting problems in astrophysics (and applied physical sciences). This course will survey fluid dynamics (including magnetohydrodynamics and some plasma physics -- time permitting) important for understanding various astronomical and terrestrial phenomena. Topics include basic fluid and MHD concepts and equations, waves and instabilities of various types (e.g., sound, gravity, Rossby, hydromagnetic, spiral density waves; Rayleigh-Taylor, thermal, Jeans, rotational, magnetorotational instabilities), shear and viscous flows, turbulence, shocks and blast waves, etc. These topics will be discussed in different astrophysical contexts and applications, such as atmosphere and ocean, star and planet formation, stellar oscillation/rotation/magnetism, compact objects, interstellar medium, galaxies and clusters. This course is intended mainly for graduate students (both theory and observation) interested in astrophysics and space physics. Students in other areas of applied science and engineering may find the broad astrophysical and terrestrial applications useful. Well-prepared undergradate students may also take the course. No previous exposure to fluid dynamics and astronomy is required.

The students should be familiar with classical mechanics and electrodynamics at the intermediate (junior) level, and should be comfortable with vector calculus (e.g. divergence and curl of a vector).

Organization:

Weekly lectures. There will be about 6-8 problem sets. No final exam (the last problem set may serve as take-home final exam). There may be a student project during the second half of the semester (TBD). Grades will be determined by these HWs, project and participations in class. Either Letter or S/U grade option is possible. (S = attend lectures and do 70% of HWs with passing grades)

Policy statement: You should abide by the CU Code of Academic Integrity. You are encouraged to discuss homework with other students but not to collaborate on writing up the notebook solutions (and not to copy). Anything that you turn in should be your own work (homework, project). If you need any academic accommodations please register with Student Disability Services at the beginning of the term and bring me the description of the appropriate accommodations.

Recommended Books: We will not follow any book too closely, especially when it comes to astrophysical applications.

The Physics of Astrophysics II: Gas Dynamics by Frank Shu
Fluid Mechanics: An Introduction by Michel Rieutord
Applications of Classical Physics by Kip Thorne and Roger Blandford
Fluid Mechanics by Landau & Lifshitz
Hydrodynamic and Hydromagnetic Stability by S. Chandrasekhar
Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena by Zeldovich and Raizer
Plasma Physics for Astrophysics by Russell Kulsrud
Astrophysical Flows by Jim Pringle & Andrew King
The Physics of Fluids and Plasmas by Arnab Raichoudhuri
Astrophysical Fluid Dynamics Lecture Note by Gordon Ogilvie
Theory of Planetary Atmospheres by Chamberlain and Hunten

There are many other nice (famous) fluid dynamics books, many containing interesting applications (not necessarily astrophysical), such as

"Physical Fluid Dynamics" by Tritton

"An Introduction to Fluid Dynamics" by Batchelor

"Waves in Fluids" by Lighthill

"Elementary Fluid Dynamics" by Acheson

"Nonlinear Hydrodynamic Modeling: A Mathematical Introduction" ed. Hampton N. Shirer
(low-order modeling of Rayleigh-Bénard convection)

Fluid Mechanics Films

Topics covered in class: (suggested reading from Shu, LL=Landau & Lifshitz, TB=Thorne-Blandford, or other sources).

Here are the topics covered in Spring 2023 . We will cover some of the topics, and may add some new topics

8/26 (DL): Basic assumptions of Fluid Dynamics: collisions in gas and plasma. Qualitative discussion of collisionless plasma (gyro-radius condition). Basic fluid equations: mass conservation; Euler equation, and review of gravity term, magnetic force. Reading: Shu: Chapter 1; p.45-46; skim Chap.5 (You have learned that in Stellar Structure). LL: P.1-7.
8/28 (DL): Navier-Stokes eqn (just give the viscous force but not deriving it). Momentum equation in conservative form; EOS and energy equation. Barotropic flows (examples of barotropic relations: adiabatic vs isentropic flows, ISM, degenerate stars, etc). Sound wave (derived). Reading: LL: sections 1,2,3,7. Shu: p.44-46, p.64, p.107; BT: Skim through section 13.1-13.6.
8/30 (DL): Incompressible flow (why/when valid?). Properties of inviscid barotropic flow: vorticity equation and Bernoulli's theorem. Interpretation of vorticity. Applications: irrotational flows, coalescing NS binaries, Magnus force. Reading: Shu: Chap.6. LL: p.8-9, p.12-18.
9/4 (DL): Bondi flow (estimate and formal derivation). Bondi-Lyttleton accretion. Sub-sonic solution (relevance to giant planet formation; qualitative discussion). Parker stellar wind solution. Reading: Shu: Chap.6. LL: p.8-9, p.12-18.
9/9: No class (makeup class was on 8/30)
9/11 (Steve): Viscous flows: 1D equation, viscous stress tensor, shear and bulk viscosities. Microphysics of viscosity. Scaling and Reynolds number. Reading: LL: section 15 to Eq. (15.10); section 19 to Eq. (19.2).
9/16 (Steve): Videos of flows past a cylinder and a sphere at different Re. Dynamical similarity. Flow passing a sphere: low-Re flow (Stokes flow); behavior as a function of Re. Viscous boundary layer (estimate), drag force in the presence of BL. General drag force formula (Epstein, Stokes, etc), application to dust dynamics in pre-solar nebula. Reading: Tritton: Chap.3, Chap.8.
9/18 (DL): Sound wave generation: oscillating ball, monopolar radiation, higher-order radiation. Lighthill's law. Reading: Shu: Chap.6. For those interested in giant planet formation: Section III.C of Armitage review. Parker wind. Sound wave is discussed thoroughly in Chap.8 of LL. See also relevant chapters in Thorne-Blandford
9/23 (DL): Sound wave with gravity, Jean's instability. Isothermal cloud and Jean Mass (digression on Virial theorem). Gravity waves (surface waves on a pond). Eulerian vs Lagrangian perturbation.
9/25 (DL): Gravity waves continued: Eulerian vs Lagrangian perturbation. Derive the dispersion relation. Order-of-magnitude discussion on deep-water and shallow-water waves, wave refraction and Tsunamis.
9/30 (DL): Nonlinear shallow water wave equations. Method of characteristics applied to propagating waves, Burger's equation. Physical discussion of wave steepening. Rayleigh-Taylor instability (and application to SN explosion). Reading: Thorne-Blandford: Sections 16.2-16.3
10/2 (DL): Waves in plane-parallel atmospheres: derive local dispersion relation, sound waves vs gravity waves, Physics of Brunt-Vasala frequency. Convective instability (Schwarzschild criterion). Reading: Pringle-King, Chap.5.
10/7 (Steve): Kelvin-Helmholtz instability: images of jets from Cen A, M87, HH111; time-lapse animation of Jupiter cloud bands. Overview of basic methods of analysis used in fluid dynamics. Derivation of Kelvin-Helmholtz in 2D with gravity stabilization. Excitation of ocean waves by wind; physical interpretation. Reading: Chandrasekhar, sections 100-101. LL, section 30.
10/9 (Steve): Video explaining Kelvin-Helmholtz waves in clouds. Shear flow instability: Rayleigh inflexion point theorem. Competition between shear and stable stratification: Richardson criterion. Convective instability including dissipation: Rayleigh-Bénard convection. Reading: Shu, Chap.8 (pp.93-105). Pringle-King: 10.1-10.3, 10.8. Chandrasekhar: sections 5-12, 15.
10/16 (Steve): Images of nonlinear dynamics: secondary Kelvin-Helmholtz instability on top of Rayleigh-Taylor, plus chaotic trajectories in the Lorenz model and their sensitivity to initial conditions. Rayleigh-Bénard convection continued: initial instability and critical Rayleigh number, Lorenz's low-order model, nonlinear saturation, bifurcations, possibility of chaos. Reading: Shirer, Chap.2.
10/21 (Steve): Convection conclusion: video of solar granulation. Boussinesq approximation. Mixing length theory of convection. Rotation: Rayleigh's criterion for rotational instability. Fluid dynamics in rotating frame. Rotational distortion. Reading: Tritton, appendix to Chap.14. Shu, Chap.10.
10/23 (Steve): Rotation and Coriolis force: images of solar internal rotation, weather maps showing isobars and wind animation. Rossby number. Geostrophic flows. Taylor-Proudman theorem. Rossby waves. Quick overview of global Rossby waves and the CFS instability. Beta plane approximation. Reading: Tritton, Chap.16 (2nd ed.). Chamberlain and Hunten, p. 88.
10/28 (Steve): Images of p-, g-modes and frequencies in helioseismology from Berkeley and Yale. Kelvin's circulation theorem in a rotating frame. Explanation of Rossby waves based on theorem. Correspondence of eigenmodes, eigenvalues for local vs. global Rossby waves. Rotational splitting of resonant mode frequencies in stellar interior (p-, g-modes, etc.). Helioseismology as a means of inferring differential rotation in the Sun. Ekman layer intro and estimate of width. Reading: Chandrasekhar, section 22. Tritton, Chap.16 (2nd ed.).
10/30 (Steve): Videos of turbulence developing in a pipe (Poiseuille flow) and a planar boundary layer. Definitions of streamlines, streaklines (dye injection), pathlines (tracer particles). Ekman spiral. Gravity-inertial waves in rotating fluid (and barotropic inertial waves). Turbulence: intro, description, definitions. Reading: Tritton, sections 6.1, 16.5 (2nd ed.). Shu, Chap.9.
11/4 (Steve): Video of transition to turbulence in a wake. Kolmogorov theory (derive spectrum). 3D and 2D turbulence: conservation laws, energy and enstrophy cascade (mention Kraichnan theory). Reading: Shu, Chap.9.
11/6 (Steve): Nonlinear sound wave steepening. Shock waves: Shock jump condition. Blast waves: Sedov-Taylor solution. Reading: Shu, Chaps.15,17.
11/11 (DL): Blast waves: Simple analysis, Sedov-Taylor solution. Forward an reverse shock, evolution of supernova remnants. Reading: Shu, Chap.15
11/13 (DL): Stellar oscillations: brief observation background. Stellar oscillations theory: Radial pulsation: equations, estimate of discrete modes. Radial pulsational instability of massive stars.
11/18 (DL): Nonradial pulsations: derive eqns in convenient forms, boundary conditions. WKB dispersion relation. propagation diagram, mode classification. Gravity waves and convection: Composition gradients and Ledoux criterion. Double diffusive instabilities, salt fingers. Nonadiabatic effect and mode excitation (intro): energy equations. Nonadiabatic effect and mode excitation (kappa and epsilon mechanisms). Application of stellar oscillations.
11/20 (DL): Accretion disk intro. Basic equations for axisymmetric thin disks. Boundary layers; origin of viscosity, disk turbulence. Reading: Shu: Chap.7.
11/25 (DL): Waves in disks: dispersion relation, Toomre criterion, Safraonov-Goldrecih-Ward instability. Spiral density waves: kinematics. (wave propgation zone, Q-barrier, corotation and Lindblad resonances. Negative energy waves, super-reflection and corotation amplifier.) Reading: Shu, Chap.12,13
12/2 (Steve): MHD equations: magnetic forces (pressure and tension), magnetic evolution (diffusion, flux freezing). Amplification of toroidal fields in the Sun. Reading: Shu, Chap.21.
12/4 (Steve): Hydromagnetic waves: Alfvén and fast/slow magnetosonic modes. Magnetostatic equilibrium: pressure-balanced plasma column, sausage and kink instabilities. Twist stabilization in Z-pinch and tokamak geometries. Relation of ExB drifts to MHD fluid velocity. Reading: Shu, Chap.22.
12/9 (Steve): Magnetic cloud, virial theorem. Critical mass to flux ratio (magnetic flux problem of star formation). Ambipolar diffusion. Qualitative discussion of torsional Alfven waves and magnetic braking. Plasma beta. Magnetic buoyancy (Parker's idea). Babcock's model of the solar cycle. Reading: Shu, Chaps.24,27,pp.352-353.

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